Let $X$ be a smooth projective algebraic variety over an algebraically closed field $k$. If $k=\mathbb{C}$, we know by work of Kuranishi that the base of the versal deformation of $X$ is the germ at $0$ of the fiber over $0$ of a holomorphic map $K:H^1(X, T_X)\to H^2(X, T_X)$ (defined in the neighborhood of 0), called the Kuranishi map.

This means that, if $S_i$ are the power series rings associated to $H^i(X, T_X)$ (i.e., the completions of $Sym^* H^i(X, T_X)^*$), there is a map $k: S_2 \to S_1$ such that $R = S_1 \otimes_{S_2} k$ pro-represents the deformation functor of $X$.

Question. Can one construct the map $k$ using algebraic methods?

Probably one should assume that $k$ is of characteristic zero (or replace power series rings by completed divided power algebras...).

If $L^\bullet$ is the dgla (differential graded Lie algebra) governing your deformation problem (the Kodaira-Spencer dgla $\oplus A^{0,p}(X,T_X)$ in your example), the formal Kuranishi theorem states that
for every splitting $\delta$ of $L^\bullet$, there exists a hull $Kur^\delta_{L^\bullet}\to Def_{L^\bullet}$, the formal Kuranishi space. Here
$Def_{L^\bullet}:Art_{k}\to Sets$ is the deformation functor associated to $L^\bullet$.
A splitting of $L^\bullet$ is a degree $-1$ linear map, $\delta:L^\bullet\to L^\bullet[-1]$, such that $\delta^2=0$,
$d\delta d=d$, $\delta d\delta=\delta$. It plays the role of $d^\ast G$ in Hodge theory,
where $G$ is Green's operator and $d^\ast$ is the adjoint of $d$.

Manetti's article that Francesco mentions is a great introduction. Another reference that I am very fond of is Goldman and Millson's paper The Homotopy invariance of Kuranishi space,
where they compare the algebraic and analytic description of the hull.